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Re: distribution assumption of Eta in NONMEM

From: Nick Holford <n.holford>
Date: Mon, 31 May 2010 22:43:10 +0200

Leonid,

The result is what I expected. NONMEM just estimates the variance of the
random effects. It doesn't promise to tell you anything about the
distribution.

It is indeed bad news for simulation if your simulation relies heavily
on the assumption of a normal distribution and the true distribution is
quite different.

I think you have to be very careful looking at posthoc ETAs. They are
not informative about the true ETA distribution unless you can be sure
that you have low shrinkage. If shrinkage is not low then a true uniform
will become more normal looking because the tails will collapse.

The approach that Mats seems to suggest is to try different
transformations of NONMEM's ETA variables to try to lower the OFV. What
is not clear to me is why these transformations which lower the OFV will
make the simulation better when the ETA variables that are used for the
simulation are required to be normally distributed.

Imagine I use this for estimation:
CL=POPCL*EXP(ETA(1)) where the true ETA is uniform
If I now use the estimated OMEGA(1,1) which will be a good estimate of
the uniform distribution variance, uvar, for simulation then I am using
CL=POPCL*EXP(N(0,uvar))
which will be wrong because I am now assuming a normal distribution but
using the variance of a uniform.

Now suppose I try:
CL=POPCL*TRANS(ETA(1)) where TRANS is some transformation that lowers
the OFV to the lowest I can find but the true ETA is still uniform
If I now use the same transformation for simulation with an OMEGA(1,1)
estimate of the variance transvar
CL=POPCL*TRANS(N(0,transvar)) which uses a normal distribution then why
should I expect the simulated distribution of CL to resemble the true
distribution with a uniform ETA?

Nick

Leonid Gibiansky wrote:
> Hi Nick,
> I think, I understood it from your original e-mail, but it was so
> unexpected that I asked to confirm it.
>
> Actually, not a good news from your example.
>
> Nonmem cannot distinguish two models:
> with normal distribution, and
> with uniform distributions
> as long as they have the same variance.
>
> So if you simulate from the model, you will end up with very different
> results: either simular to the original data (if by chance, your
> original problem happens to be with normal distribution) or very
> different (if original distribution was uniform).
>
> This shows the need to investigate normality of posthoc ETAs very
> carefully.
>
> Very interesting example
> Thanks
> Leonid
>
> --------------------------------------
> Leonid Gibiansky, Ph.D.
> President, QuantPharm LLC
> web: www.quantpharm.com
> e-mail: LGibiansky at quantpharm.com
> tel: (301) 767 5566
>
>
>
>
> Nick Holford wrote:
>> Leonid,
>>
>> I meant by OMEGA(1) the OMEGA value estimated by NONMEM. I suppose I
>> should have written OMEGA(1,1) to be more precise -- sorry!
>>
>> Nick
>>
>> Leonid Gibiansky wrote:
>>> Nick, Mats
>>>
>>> I would guess that nonmem should inflate variance (for this example)
>>> trying to fit the observed uniform (-0.5, 0.5) into some normal N(0,
>>> ?). This example (if I read it correctly) shows that Nonmem somehow
>>> estimates variance without making distribution assumption.
>>> Nick, you mentioned:
>>>
>>> "the mean estimate of OMEGA(1) was 0.0827"
>>>
>>> does it mean that Nonmem-estimated OMEGA was close to 0.0827 or you
>>> refer to the variances of estimated ETAs?
>>>
>>> Thanks
>>> Leonid
>>>
>>>
>>> --------------------------------------
>>> Leonid Gibiansky, Ph.D.
>>> President, QuantPharm LLC
>>> web: www.quantpharm.com
>>> e-mail: LGibiansky at quantpharm.com
>>> tel: (301) 767 5566
>>>
>>>
>>>
>>>
>>> Mats Karlsson wrote:
>>>> Nick,
>>>>
>>>>
>>>>
>>>> It has been showed over and over again that empirical Bayes
>>>> estimates, when individual data is rich, will resemble the true
>>>> individual parameter regardless of the underlying distribution.
>>>> Therefore I don’t understand what you think this exercise contributes.
>>>>
>>>>
>>>>
>>>> Best regards,
>>>>
>>>> Mats
>>>>
>>>>
>>>>
>>>> Mats Karlsson, PhD
>>>>
>>>> Professor of Pharmacometrics
>>>>
>>>> Dept of Pharmaceutical Biosciences
>>>>
>>>> Uppsala University
>>>>
>>>> Box 591
>>>>
>>>> 751 24 Uppsala Sweden
>>>>
>>>> phone: +46 18 4714105
>>>>
>>>> fax: +46 18 471 4003
>>>>
>>>>
>>>>
>>>> *From:* owner-nmusers
>>>> [mailto:owner-nmusers
>>>> *Sent:* Monday, May 31, 2010 6:05 PM
>>>> *To:* nmusers
>>>> *Cc:* 'Marc Lavielle'
>>>> *Subject:* Re: [NMusers] distribution assumption of Eta in NONMEM
>>>>
>>>>
>>>>
>>>> Hi,
>>>>
>>>> I tried to see with brute force how well NONMEM can produce an
>>>> empirical Bayes estimate when the ETA used for simulation is
>>>> uniform. I attempted to stress NONMEM with a non-linear problem
>>>> (the average DV is 0.62). The mean estimate of OMEGA(1) was 0.0827
>>>> compared with the theoretical value of 0.0833.
>>>>
>>>> The distribution of 1000 EBEs of ETA(1) looked much more uniform
>>>> than normal.
>>>> Thus FOCE show no evidence of normality being imposed on the EBEs.
>>>>
>>>> $PROB EBE
>>>> $INPUT ID DV UNIETA
>>>> $DATA uni1.csv ; 100 subjects with 1 obs each
>>>> $THETA 5 ; HILL
>>>> $OMEGA 0.083333333 ; PPV_HILL = 1/12
>>>> $SIGMA 0.000001 FIX ; EPS1
>>>>
>>>> $SIM (1234) (5678 UNIFORM) NSUB=10
>>>> $EST METHOD=COND MAX=9990 SIG=3
>>>> $PRED
>>>> IF (ICALL.EQ.4) THEN
>>>> IF (NEWIND.LE.1) THEN
>>>> CALL RANDOM(2,R)
>>>> UNIETA=R-0.5 ; U(-0.5,0.5) mean=0, variance=1/12
>>>> HILL=THETA(1)*EXP(UNIETA)
>>>> Y=1.1**HILL/(1.1**HILL+1)
>>>> ENDIF
>>>> ELSE
>>>>
>>>> HILL=THETA(1)*EXP(ETA(1))
>>>> Y=1.1**HILL/(1.1**HILL+1) + EPS(1)
>>>> ENDIF
>>>>
>>>> REP=IREP
>>>>
>>>> $TABLE ID REP HILL UNIETA ETA(1) Y
>>>> ONEHEADER NOPRINT FILE=uni.fit
>>>>
>>>> I realized after a bit more thought that my suggestion to transform
>>>> the eta value for estimation wasn't rational so please ignore that
>>>> senior moment in my earlier email on this topic.
>>>>
>>>> Nick
>>>>
>>>>
>>>> --
>>>>
>>>> Nick Holford, Professor Clinical Pharmacology
>>>>
>>>> Dept Pharmacology & Clinical Pharmacology
>>>>
>>>> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New
>>>> Zealand
>>>>
>>>> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
>>>>
>>>> email: n.holford
>>>>
>>>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>>>>
>>
>> --
>> Nick Holford, Professor Clinical Pharmacology
>> Dept Pharmacology & Clinical Pharmacology
>> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
>> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
>> email: n.holford
>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>>

--
Nick Holford, Professor Clinical Pharmacology
Dept Pharmacology & Clinical Pharmacology
University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New Zealand
tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
email: n.holford
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford


Received on Mon May 31 2010 - 16:43:10 EDT

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